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In this lesson, we will learn how to find the velocity and displacement using definite integration.

Q1:

A particle is moving in a straight line such that its speed at time π‘ seconds is given by Given that its initial position π = 1 6 0 m , find its position when π‘ = 3 s e c o n d s .

Q2:

A particle started moving in a straight line from the origin such that its acceleration at time π‘ seconds is given by Given that its initial velocity was 14 m/s, determine its velocity π£ and its displacement π when π‘ = 2 s e c o n d s .

Q3:

The diagram below shows the acceleration of a particle which was initially at rest. What was its velocity at π‘ = 7 s ?

Q4:

The acceleration-time graph of a particle which was initially at rest is shown below. What was its velocity at π‘ = 1 1 s ?

Q5:

A particle is moving in a straight line such that its velocity at time π‘ seconds is given by Given that its initial position π = 1 3 0 m , find an expression for its position at time π‘ seconds.

Q6:

A particle started moving in a straight line from point towards point . Its velocity after seconds is given by After 2 seconds, another particle started moving in a straight line from point towards point . This particle was accelerating at 0.9 m/s^{2}. The two particles collided 6 seconds after the first particle started moving. Find the distance .

Q7:

A particle is moving in a straight line such that its velocity at time π‘ seconds is given by Given that when π‘ = π 2 seconds its position from a fixed point is π = 2 0 3 m , find an expression for π in terms of π‘ .

Q8:

A particle started moving in a straight line. Its acceleration at time π‘ seconds is given by Find the maximum velocity of the particle π£ m a x and the distance π₯ it travelled before it attained this velocity.

Q9:

A particle started moving from rest along the π₯ -axis from a point at π₯ = 1 0 m . Its acceleration at time π‘ seconds is given by Express its velocity π£ and its displacement π₯ after time π‘ seconds.

Q10:

A particle, starting from rest, began moving in a straight line. Its acceleration π , measured in metres per second, and the distance π₯ from its starting point, measured in metres, satisfy the following equation π = π₯ 1 5 2 . Find the speed π£ of the particle when π₯ = 1 1 m .

Q11:

A particle is moving in a straight line such that its acceleration at time π‘ seconds is given by Given that its initial velocity is 1 2 1 3 m/s and its initial position from a fixed point is 7 1 0 m, determine its position when π‘ = 2 π seconds.

Q12:

The figure shows a velocity-time graph for a particle moving in a straight line. Find the magnitude of the displacement of the particle.

Q13:

A particle is moving in a straight line such that its velocity after π‘ seconds is given by Find the distance covered during the time interval between π‘ = 0 s and π‘ = 1 2 s .

Q14:

The acceleration of a particle moving in a straight line, at time π‘ seconds, is given by When π‘ > 1 3 , the particle moves with uniform velocity π£ . Determine the velocity π£ and the distance, π , covered by the particle in the first 23 s of motion.

Q15:

A body started moving along the π₯ -axis from the origin at an initial speed of 10 m/s. When it was π meters away from the origin and moving at π£ m/s, its acceleration was ( 4 5 π ) β π m/s^{2} in the direction of increasing π₯ . Determine π when π£ = 1 1 / m s .

Q16:

A body moves in a straight line. At time π‘ seconds, its acceleration is given by Given that the initial displacement of the body is 9 m, and when π‘ = 2 s , its velocity is 27 m/s, what is its displacement when π‘ = 3 s ?

Q17:

A particle is moving in a straight line such that its acceleration, π metres per second squared, and displacement, π₯ metres, satisfy the equation π = 2 6 π β π₯ . Given that the particle's velocity was 12 m/s when its displacement was 0 m, find an expression for π£ 2 in terms of π₯ , and determine the speed π£ m a x that the particle approaches as its displacement increases.

Q18:

A particle moves in a straight line. At time π‘ seconds, its velocity, in meters per second, is given by What is its displacement in the interval 0 β€ π‘ β€ π 2 s e c o n d s ?

Q19:

A particle started moving in a straight line such that its acceleration, measured in metres per second, and its displacement from a fixed point, measured in metres, satisfy the following equation: π = π₯ β 6 . Given that its initial displacement from the fixed point was 9 metres and its initial velocity was 5 m/s, determine its velocity when π = 0 .

Q20:

A particle moves along the π₯ axis. It is initially at rest at the origin. At time π‘ seconds, the particleβs acceleration is given by How long does it take for the particle to return to the origin?

Q21:

A particle started moving from a fixed point in a straight line such that its acceleration π , measured in metres per second squared, and its position π₯ , measured in metres, satisfy the following equation: π = 2 π₯ + 2 4 . Given that the initial velocity of the particle was 2 m/s, find an expression for π£ 2 in terms of π₯ .

Q22:

A particle starts from rest and moves in a straight line. At time π‘ seconds, its velocity is given by How long does it take for the particle to return to its starting point?

Q23:

The given acceleration-time graph represents the motion of a particle. Given that the particle moved in a straight line and was at rest at time π‘ = 0 s , find the velocity of the particle at π‘ = 1 2 s .

Q24:

A particle started moving on the π₯ -axis from the origin π at an initial velocity π£ 0 . At time π‘ seconds, the particleβs acceleration is given by Given that after 6 seconds, the particleβs displacement was 75 m in the right from point π , Determine π£ 0 . If the body passed by π once again, determine its velocity π£ at this moment.

Q25:

Two bodies, initially at rest at the same point, start to move in the same direction along the same straight line. At time π‘ seconds ( π‘ β₯ 0 ) , their velocities are given by π£ = ( 2 3 π‘ β 5 8 8 ) / 1 c m s and π£ = οΉ 3 π‘ β 6 1 π‘ ο / 2 2 c m s . Determine the time taken for the two bodies to be 2β744 cm apart.

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